Notes: `(dr)/(ds) = 0.75r` Find the general solution of the differential equation

Monday, 18 November 2013

$\displaystyle\frac{{{d}{r}}}{{{d}{s}}}={0.75}{r}$ Find the general solution of the differential equation

The general solution of a differential equation in a form of can be evaluated using direct integration. The derivative of y denoted as $\displaystyle{y}'$ can be written as $\displaystyle\frac{{{\left.{d}{y}\right.}}}{{{\left.{d}{x}\right.}}}$ then $\displaystyle{y}'={f{{\left({x}\right)}}}$ can be expressed as $\displaystyle\frac{{{\left.{d}{y}\right.}}}{{{\left.{d}{x}\right.}}}={f{{\left({x}\right)}}}$

For the problem $\displaystyle\frac{{{d}{r}}}{{{d}{s}}}={0.75}{r}$ , we may apply variable separable differential equation in which we set it up as $\displaystyle{f{{\left({y}\right)}}}{\left.{d}{y}\right.}={f{{\left({x}\right)}}}{\left.{d}{x}\right.}$ .

Then, $\displaystyle\frac{{{d}{r}}}{{{d}{s}}}={0.75}{r}$ can be rearrange into $\displaystyle\frac{{{d}{r}}}{{r}}={0.75}{d}{s}$ .

Applying direct integration on both sides:

$\displaystyle\int\ldots$

Then, $\displaystyle\frac{{{d}{r}}}{{{d}{s}}}={0.75}{r}$ can be rearrange into $\displaystyle\frac{{{d}{r}}}{{r}}={0.75}{d}{s}$ .

Applying direct integration on both sides:

$\displaystyle\int\frac{{{d}{r}}}{{r}}=\int{0.75}{d}{s}$ .

For the left side, we apply the basic integration formula for logarithm: $\displaystyle\int\frac{{{d}{u}}}{{u}}={\ln}{\left|{u}\right|}+{C}$

$\displaystyle\int\frac{{{d}{r}}}{{r}}={\ln}{\left|{r}\right|}$

For the right side, we may apply the basic integration property: $\displaystyle\int{c}\cdot{f{{\left({x}\right)}}}{\left.{d}{x}\right.}={c}\int{f{{\left({x}\right)}}}{\left.{d}{x}\right.}$ .